V. Demenko.

Mechanics of Materials 1

LECTURE 7. Fundamentals of the Theory of Stress

1. State of Stress (Stress State) at a Point
A set of stresses occurring on all planes passing through the point in question is
called the state of stress at the point.

Fig.1
Suppose there is a certain body subjected to an arbitrary force system
(Fig.1). We assume the state of stress varies sufficiently slowly when passing
from point to point and it is always possible to choose a sufficiently small area
for which the state of stress might be considered as homogeneous in the vicinity
of point A. It is apparent that such an approach is feasible only within the

Fig.2
hypothesis of continuous medium. It allows passing to infinitely small volume.
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 V. Demenko. Mechanics of Materials 2

We isolate an elementary volume in the right parallelepiped form of the

vicinity point, with six sections. There is a question:
If the dimensions of the parallelepiped are reduced, it will contract into
that point. At the limiting all faces of the parallelepiped pass through point A,
and the stresses on the corresponding cutting planes may be regarded as stresses
at the point concerned.
The total stress acting on a cutting plane can be resolved into three
components: one along the normal respectively to the plane and two in the
sectional plane.
We will denote the normal stress with σ having a subscript
corresponding to the appropriate axis (x, y or z). The shear stress will be
denoted with the symbol τ with two subscripts: the first corresponds to the
axis perpendicular to the plane, and the second – to the axis the vector is
directed along.
The stresses acting on three faces of the element (on three mutually
perpendicular planes passing through the point) are shown in Fig.2. The same
stresses but opposite in sense occur on the hidden faces of the element.
2. Law of Equality of Shearing Stresses
The system of forces applied to the element must satisfy the conditions of
equilibrium. Since the forces acting on the opposite faces are of different sign,
the first three conditions of equilibrium are identically satisfied and the sums of
all forces projections on the x, y and z axes are zero. It remains to check whether
the sums of all forces moments vanish with respect to the x, y and z axes.
The zero sum of the moments for x axis is fulfilled if the moment of the
force τyz dxdz is equal to the moment of the force τzy dxdy , i.e.
∑ M x ( F ) = 0 → (τ zy dxdy )dz − (τ yz dxdz )dy = 0 , τ zy = τ yz ;

∑ M y ( F ) = 0 → (τ zx dxdy ) dz − (τ xz dydz ) = 0 , τ zx = τ xz ;

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 V. Demenko. Mechanics of Materials 3

∑ M ( F ) = 0 → (τ
z xy dydz )dx − (τ yx dxdz ) = 0 , τ xy = τ yz . (1)
τ xy = τ yx , τ yz = τ zy , τ zx = τ xz .

Thus, on two planes being at right angles to each other, the components of
shearing stresses perpendicular to the common edge are equal and directed
either both toward the edge or both away from the edge. This is the law of
equality of shearing stresses.

3.Determination of Stresses on a Plane of General Position

G: σ x, σ y, σ z, τ xy, τ yz, τ zx (i.e. six stress components are given on three

orthogonal planes).
It is required to determine the stresses passing through the given point on
any plane. We isolate an elementary volume from a stressed body in the vicinity
of point O. Three faces of the isolated element coincide with the co-ordinate
planes of the system x, y, z. The fourth face is formed by a cutting plane of

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 V. Demenko. Mechanics of Materials 4

general position. Its orientation in space will be defined by the direction cosines
of the normal ν in the system of axes x, y, z, i.e. by quantities l, m, n.
We will project the total stress vector on the ABC plane of general position

located on the x, y and z axes. Let these projections be denoted by Fx, Fy, Fz
respectively.
Let the areas of the triangles ABC, OBC, OAC, OAB be denoted by A, Ax,

Ay, Az respectively. Obviously Ax =A⋅ l, Ay =A⋅ m, Az =A⋅ nl where l, m and n

are the direction cosines of the normal ν .
By projecting all forces acting on the element of x, y and z axes we obtain
Fx ⋅ A = σ x Ax +τ yx Ay +τ zx ⋅ Az ,

Fy ⋅ A = τ xy Ax + σ x Ay +τ zy ⋅ Az , (2)
Fz ⋅ A = τ xz Ax +τ yz Ay + σ z ⋅ Az .

or in accordance with relations (2)

Fx = σ x ⋅ l +τ yx m +τ zx n ,

Fy = τ xy ⋅ l +σ y m +τ zy n , (3)
Fz = τ xz ⋅ l +τ yz m +σ z n .

The total, normal and shearing stresses on any plane passing through the
point in question can easily be determined using formulas
Fν = Fx2 + Fy2 + Fz2 ,
σν = Fx ⋅ l + Fy m + Fz n , (4)
τν = Fν2 − σν2 .

Thus, indeed, the projections Fx, Fy and Fz for any plane defined by

means of direction cosines l, m and n are expressed in terms of six basic σ x,

σ y, σ z, τ xy, τ yz, τ zx. In another words, the state of stress at a point is defined
by six components of stress.
The state of stresses represents a tensor
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 V. Demenko. Mechanics of Materials 5

 σx τ yx τ zx 
 
Tσ =  τ xy σy τ zy  (5)
 
 τ xz τ yz σz 

4.Principal Planes. Principal Stresses

At any chosen point of stressed body exists such a system of axes x, y, z

where the shearing stresses τ xy, τ yz and τ zx are zero. These axes are called
principal axes.
The corresponding, mutually perpendicular planes are called principal
planes, and the normal stresses on them are called principal stresses. These

stresses are denoted with σ xy, σ yz and σ zx in the magnitude increase order

σ 1 ≥ σ 2 ≥ σ 3. (6)

It is algebraic relation.
According to the principal stresses it is usual to distinguish:
1. Uniaxial stress state

σ1 ≠ 0 , σ 2 = 0 , σ3 = 0 ; σ1 = 0 , σ 2 = 0 , σ3 ≠ 0 .

2. Biaxial stress state

3. Triaxial stress state (for example):

σ1 ≠ 0 ,

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 V. Demenko. Mechanics of Materials 6
σ2 ≠ 0 ,

σ3 ≠ 0 .

VOCABULARY 7
state of stress __________________ напряженное состояние
set __________________ совокупность
in the vicinity of __________________ в окрестности
sufficient __________________ достаточный
apparent __________________ очевидный
feasible __________________ возможный
infinitely small volume __________________ бесконечно малый объем
parallelepiped __________________ параллелепипед
reduce __________________ сокращать, уменьшать
contract __________________ сжимать(ся)
correspond __________________ соответствовать
appropriate __________________ подходящий,
соответствующий
symbol __________________ символ
orthogonal __________________ ортогональный,
прямоугольный
tensor __________________ тензор
matrix __________________ матрица
uniaxial __________________ одноосный
biaxial __________________ двухосный
triaxial __________________ трехосный

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